A Local Dimension Test for Numerically Approximated Points on Algebraic Sets

Given a numerical approximation to a point p on the set V of common zeroes of a set of multivariate polynomials with complex coefficients, this article presents an efficient method to compute the maximum dimension of the irreducible components of V which pass through p, i.e., a local dimension test. Such a test, used to filter out the so-called “junk points,” is a crucial element in the numerical irreducible decomposition algorithms of Sommese, Verschelde, and Wampler. Computational evidence presented in this article illustrates that, with this new local-dimension test, “junk-point filtering” is no longer a bottleneck in the computation of a numerical irreducible decomposition. For moderate size examples this results in well over an order of magnitude improvement in the computation of a numerical irreducible decomposition. Also, to compute the irreducible components of a fixed dimension, it is no longer necessary to compute the numerical irreducible decomposition of all higher dimensions.